How can I set my origin point in the center of my cube so it can be the same in both edit mode and object mode?
The origin of your object is where you put it .. you can still see it there, the dot in the center of the half-cube. 

What's not where you want it is the 'Pivot Point for Rotation / Scaling', which you have set to either 'Individual Origins' or 'Median Point' (It's hard to tell from the picture - just to the left of the manipulator button). That, in either case, will be at your selected vertex, if you only have one vertex selected. 

If want your transformation of elements to be centered on the object's origin while in Edit Mode, go into Object Mode, select the cube, and hit ShiftS > 'Cursor to Selected'. That will move the 3D cursor to the origin of the cube. Now, back in Edit Mode, select '3D Cursor' as the pivot for rotation/scaling, either by using keyboard . or selecting from this rollout:



In Object Mode, you can manipulate only objects, not their meshes, and their 'positions', for transformation purposes, are defined to be the positions of their origins.